Could anybody answer,please,why is it no possible to save tif files in Grayscale Mode from LR? (Or am I wrong)

Perhaps I'm misunderstanding what it is you're trying to do but there's no problem with saving a grey-scale (ie, B&W) TIFF file from Lightroom and no export to Photoshop is necessary in order to do it. I assume you're starting with a colour RAW or DNG image. Convert it to B&W in the develop module. Switch back to the library module when done. Click the Export button, lower left corner of screen. In the export dialog box select your destination folder, scroll down to the File Settings section and switch from the default jpeg to tiff using the Image Type pull-down menu. Make any other settings you need, hit the Export button in the lower-right corner of the dialog box and you're done.

Perhaps I'm misunderstanding what it is you're trying to do but there's no problem with saving a grey-scale (ie, B&W) TIFF file from Lightroom and no export to Photoshop is necessary in order to do it. I assume you're starting with a colour RAW or DNG image. Convert it to B&W in the develop module. Switch back to the library module when done. Click the Export button, lower left corner of screen. In the export dialog box select your destination folder, scroll down to the File Settings section and switch from the default jpeg to tiff using the Image Type pull-down menu. Make any other settings you need, hit the Export button in the lower-right corner of the dialog box and you're done.

The problem is that when i have finished all the editing of the photo in LR ,I want to export a TIF file in grayscale , Gray gamma 2.2 ,in order not to have a huge file.By the way ,I think that this may interest many people who may want to send "lighter" than RGB files(by mail for example) since the quality is not affected from the conversion.I still do not understand why this possibility is not included in LR.

Really, I've given you the reason - there is little demand for it. There's much more demand for CMYK, for example. Those who really need Grayscale Mode are usually those who also have Photoshop and can create an action or droplet:

In PS, create a new document and record an action with one step - Image > Mode > Grayscale

In PS, File > Automate > Create Droplet

Point it to your action

In Destination, choose Save and Close

Save the Droplet to your desktop

In Lightroom, Preferences, Presets, Show LR Presets folder

Should now be in Finder/Explorer, so go into this Lightroom folder and Export Actions

Move the Droplet into this folder

Restart LR

Export a file and at the bottom of the export dialog notice the Post Processing section

From its dropdown list, choose your Droplet and click OK

Once the file has exported, PS should open it, make it Grayscale Mode, and save and close it

Really, I've given you the reason - there is little demand for it. There's much more demand for CMYK, for example. Those who really need Grayscale Mode are usually those who also have Photoshop and can create an action or droplet:

In PS, create a new document and record an action with one step - Image > Mode > Grayscale

In PS, File > Automate > Create Droplet

Point it to your action

In Destination, choose Save and Close

Save the Droplet to your desktop

In Lightroom, Preferences, Presets, Show LR Presets folder

Should now be in Finder/Explorer, so go into this Lightroom folder and Export Actions

Move the Droplet into this folder

Restart LR

Export a file and at the bottom of the export dialog notice the Post Processing section

From its dropdown list, choose your Droplet and click OK

Once the file has exported, PS should open it, make it Grayscale Mode, and save and close it